Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $a \neq 0$. $q = \dfrac{16a + 40}{4} \div \dfrac{a(2a + 5)}{9} $
Dividing by an expression is the same as multiplying by its inverse. $q = \dfrac{16a + 40}{4} \times \dfrac{9}{a(2a + 5)} $ When multiplying fractions, we multiply the numerators and the denominators. $q = \dfrac{ (16a + 40) \times 9 } { 4 \times a(2a + 5) } $ $ q = \dfrac {9 \times 8(2a + 5)} {4 \times a(2a + 5)} $ $ q = \dfrac{72(2a + 5)}{4a(2a + 5)} $ We can cancel the $2a + 5$ so long as $2a + 5 \neq 0$ Therefore $a \neq -\dfrac{5}{2}$ $q = \dfrac{72 \cancel{(2a + 5})}{4a \cancel{(2a + 5)}} = \dfrac{72}{4a} = \dfrac{18}{a} $